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Properties

Label 16.8.27583540507...0625.3
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 29^{8}\cdot 109^{4}$
Root discriminant $38.91$
Ramified primes $5, 29, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2.C_2\wr C_2^2$ (as 16T394)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-179, -19420, -43609, -50375, -67682, -38767, 11705, 9612, 870, 1445, 828, -367, -128, 35, -6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 6*x^14 + 35*x^13 - 128*x^12 - 367*x^11 + 828*x^10 + 1445*x^9 + 870*x^8 + 9612*x^7 + 11705*x^6 - 38767*x^5 - 67682*x^4 - 50375*x^3 - 43609*x^2 - 19420*x - 179)
 
gp: K = bnfinit(x^16 - 2*x^15 - 6*x^14 + 35*x^13 - 128*x^12 - 367*x^11 + 828*x^10 + 1445*x^9 + 870*x^8 + 9612*x^7 + 11705*x^6 - 38767*x^5 - 67682*x^4 - 50375*x^3 - 43609*x^2 - 19420*x - 179, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 6 x^{14} + 35 x^{13} - 128 x^{12} - 367 x^{11} + 828 x^{10} + 1445 x^{9} + 870 x^{8} + 9612 x^{7} + 11705 x^{6} - 38767 x^{5} - 67682 x^{4} - 50375 x^{3} - 43609 x^{2} - 19420 x - 179 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(27583540507977079969140625=5^{8}\cdot 29^{8}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.91$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 29, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{717} a^{14} - \frac{58}{717} a^{13} - \frac{28}{239} a^{12} - \frac{245}{717} a^{11} - \frac{38}{717} a^{10} + \frac{69}{239} a^{9} - \frac{52}{717} a^{8} - \frac{346}{717} a^{7} + \frac{86}{717} a^{6} + \frac{9}{239} a^{5} - \frac{92}{239} a^{4} - \frac{305}{717} a^{3} + \frac{43}{717} a^{2} - \frac{89}{717} a - \frac{1}{239}$, $\frac{1}{6895822950345738877175790322464183} a^{15} + \frac{3764326955921189255415471227396}{6895822950345738877175790322464183} a^{14} + \frac{1082120107912217726558331530990023}{6895822950345738877175790322464183} a^{13} - \frac{41860431453704452132010406839129}{362938050018196783009252122234957} a^{12} - \frac{1631339204166580020505257707425447}{6895822950345738877175790322464183} a^{11} - \frac{1089736261799066671915167335927087}{2298607650115246292391930107488061} a^{10} - \frac{3415120517332125407406574254612838}{6895822950345738877175790322464183} a^{9} - \frac{727075196327519858128001965141205}{6895822950345738877175790322464183} a^{8} - \frac{2749132992389180797824632682357733}{6895822950345738877175790322464183} a^{7} - \frac{16392316425526817738557612398891}{120979350006065594336417374078319} a^{6} - \frac{608645981458792808015948558288030}{2298607650115246292391930107488061} a^{5} + \frac{511065923220262656018531313199510}{6895822950345738877175790322464183} a^{4} + \frac{1130471276293937629243385617324553}{2298607650115246292391930107488061} a^{3} - \frac{709637624351607378968156666990054}{2298607650115246292391930107488061} a^{2} + \frac{391780882697452977869206939843246}{6895822950345738877175790322464183} a + \frac{613882231788804858237568035352706}{6895822950345738877175790322464183}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4509554.69707 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.C_2\wr C_2^2$ (as 16T394):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 17 conjugacy class representatives for $C_2.C_2\wr C_2^2$
Character table for $C_2.C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109Data not computed